\contentsline {chapter}{\numberline {1}Introduction}{1}{chapter.1}
\contentsline {chapter}{\numberline {2}Related work}{5}{chapter.2}
\contentsline {section}{\numberline {2.1}Myerson's optimal auction design}{5}{section.2.1}
\contentsline {subsection}{\numberline {2.1.1}Game Setting and Basic definition}{5}{subsection.2.1.1}
\contentsline {subsection}{\numberline {2.1.2}Various kinds of Mechanisms along with restrictions}{7}{subsection.2.1.2}
\contentsline {subsection}{\numberline {2.1.3}optimal auction conditions}{8}{subsection.2.1.3}
\contentsline {section}{\numberline {2.2}Multidimensional Mechanisms}{9}{section.2.2}
\contentsline {subsection}{\numberline {2.2.1}Overview of this approach}{9}{subsection.2.2.1}
\contentsline {subsection}{\numberline {2.2.2}Game Setting and Basic definition}{10}{subsection.2.2.2}
\contentsline {subsection}{\numberline {2.2.3}A summary of the results}{10}{subsection.2.2.3}
\contentsline {section}{\numberline {2.3}Optimal Deterministic Auctions with Correlated Priors}{11}{section.2.3}
\contentsline {subsection}{\numberline {2.3.1}Motivation}{11}{subsection.2.3.1}
\contentsline {subsection}{\numberline {2.3.2}Game Setting and Basic definition}{11}{subsection.2.3.2}
\contentsline {subsection}{\numberline {2.3.3}A summary of the results}{12}{subsection.2.3.3}
\contentsline {section}{\numberline {2.4}Summary of this chapter}{12}{section.2.4}
\contentsline {chapter}{\numberline {3}Approximate Revenue via simple auction mechanisms with multiple items}{14}{chapter.3}
\contentsline {section}{\numberline {3.1}Notations and Preliminaries}{14}{section.3.1}
\contentsline {section}{\numberline {3.2}A first view of separate and bundled auction mechanism}{17}{section.3.2}
\contentsline {section}{\numberline {3.3}Approximation in two items case: lower bounds}{18}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}Separate mechanism}{18}{subsection.3.3.1}
\contentsline {subsubsection}{Different distribution}{18}{subsection.3.3.1}
\contentsline {subsubsection}{Identical distribution}{21}{equation.3.3.11}
\contentsline {subsection}{\numberline {3.3.2}Bundled mechanism}{24}{subsection.3.3.2}
\contentsline {subsubsection}{Different distribution}{24}{subsection.3.3.2}
\contentsline {subsubsection}{Identical distribution}{25}{equation.3.3.27}
\contentsline {section}{\numberline {3.4}Approximation in more than two items case: lower bounds}{26}{section.3.4}
\contentsline {subsection}{\numberline {3.4.1}Separate mechanism}{26}{subsection.3.4.1}
\contentsline {subsection}{\numberline {3.4.2}Bundled mechanism}{27}{subsection.3.4.2}
\contentsline {section}{\numberline {3.5}Approximation in multi-items case: upper bounds}{27}{section.3.5}
\contentsline {subsection}{\numberline {3.5.1}Separate mechanism}{28}{subsection.3.5.1}
\contentsline {subsubsection}{Two items}{28}{subsection.3.5.1}
\contentsline {subsubsection}{More than two items}{28}{subsection.3.5.1}
\contentsline {subsection}{\numberline {3.5.2}Bundled mechanism}{29}{subsection.3.5.2}
\contentsline {subsubsection}{Different distribution}{29}{subsection.3.5.2}
\contentsline {subsubsection}{Identical distribution}{29}{subsection.3.5.2}
\contentsline {section}{\numberline {3.6}Summary of this chapter}{30}{section.3.6}
\contentsline {chapter}{\numberline {4}Further study on simple auctions}{31}{chapter.4}
\contentsline {section}{\numberline {4.1}Monotonicity}{31}{section.4.1}
\contentsline {subsection}{\numberline {4.1.1}Monotonicity for one good}{31}{subsection.4.1.1}
\contentsline {subsection}{\numberline {4.1.2}Nonmonotonicity for multiple goods --- correlated example}{31}{subsection.4.1.2}
\contentsline {subsection}{\numberline {4.1.3}Nonmonotonicity for multiple goods --- independent example}{33}{subsection.4.1.3}
\contentsline {section}{\numberline {4.2}Lottery mechanism}{34}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}correlated example}{34}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}independent example}{36}{subsection.4.2.2}
\contentsline {section}{\numberline {4.3}Summary of this chapter}{37}{section.4.3}
\contentsline {chapter}{\numberline {5}Conclusion}{38}{chapter.5}
\contentsline {chapter}{References}{39}{chapter.5}
